Toric topology

We survey some results on toric topology.Comment: English translation of the Japanese article which appeared in "Sugaku" vol. 62 (2010), 386-41

Effects of diffusion rates on epidemic spreads in metapopulation networks

It is often useful to represent the infectious dynamics of mobile agents by metapopulation models. In such a model, metapopulations form a static network, and individuals migrate from one metapopulation to another. It is known that heterogeneous degree distributions of metapopulation networks decrease the epidemic threshold above which epidemic spreads can occur. We investigate the combined effect of heterogeneous degree distributions and diffusion on epidemics in metapopulation networks. We show that for arbitrary heterogeneous networks, diffusion suppresses epidemics in the sense of an increase in the epidemic threshold. On the other hand, some diffusion rates are needed to elicit epidemic spreads on a global scale. As a result of these opposing effects of diffusion, epidemic spreading near the epidemic threshold is the most pronounced at an intermediate diffusion rate. The result that diffusion can suppress epidemics contrasts with that for diffusive SIS dynamics and its variants when individuals are fixed at nodes on static networks.Comment: 4 figure

Acceleration of adiabatic transport of interacting particles and rapid manipulations of dilute Bose gas in ground state

We show a method to accelerate quantum adiabatic transport of identical spinless particles interacting with each other by developing the preceding fast-forward scaling theory formed for one-particle systems [Masuda and Nakamura, {\it Proc. R. Soc.} A {\bf 466}, 1135 (2010)]. We derive a driving potential which accelerates adiabatic dynamics of quantum systems composed of identical particles in order to obtain the final adiabatic states in any desired short time. We also exhibit an ideal rapid manipulation of dilute Bose gas in the ground state without energy excitation by using the fast-forward scaling theory